Re-cap • Modeling of Electrical Networks • Modeling of Mechanical ...

Re-cap

Rotational Mechanical System

• Modeling of Electrical Networks • Kirchoff’s Voltage Law • Kirchoff’s Current Law OR

• Modeling of Mechanical Systems (Linear + Rotational) • F=ma • Conservation of Momentum (Linear and/or Angular) • Conservation of Energy

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Laplace Transforms

Inverse Laplace Transforms

Integral Transform – Maps functions from the time domain to the frequency domain

Integral Transform – Maps functions from the frequency domain to the time domain

with Therefore, 1. 2. 3. 4.

Laplace Transforms are extremely useful when analyzing Linear Ordinary Differential Equations. 1. Transforms differential equations into algebraic eqns. 2. Allows us to be able to draw block diagrams from the differential eqns.

Convert f(t) to F(s) Analyze F(s). Convert back. Generally – this results in obtaining the time domain solution to the differential equation.

3. Trade-off we have to work in the frequency domain

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Some Interesting Functions

Partial Fraction Expansion To find the inverse Laplace Transform of some function F(s), we must have the ability to “factorize” fractions of polynomials

Now what?

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Partial Fraction Expansion

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Summary 1

Let N(s) denote the numerator, D(s) denote the denominator.

• (Inverse) Laplace Transforms time domain frequency domain differential equation algebraic equation

Always start with F(s) s. t. order(N(s)) < order(D(s)) Case 1: Roots of D(s) are Real & Distinct

• Partial Fraction Expansion factorizes “complicated” F(s) expressions -> significantly simplifies ability to take

Case 2: Roots of D(s) are Real & Repeated Case 3: Roots of D(s) are Complex or Imaginary

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A General N-Order Linear, Time Invariant, Differential Equation

Transfer Function

Given: c(t) – output r(t) – input G(s) == Transfer Function == output/input Furthermore, if we know G(s), then output = G(s)*input

Purpose: Relate a system’s output to its input 1. Easy separation of INPUT, OUTPUT, SYSTEM (PLANT) 2. Algebraic relationships (vs. differential) 3. Easy interconnection of subsystems in a

And, if we can take

MATHEMATICAL framework

, we get ….

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Putting it all together ….

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Underlying Assumptions

Example:

Linearity 1. Superposition System

2. Homogeneity System Because the Laplace Transform is Linear! 11

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State-Space Representation

Example 1

• Alternate approach to the Transfer Function • Time domain (vs. frequency domain) • Allows for multiple inputs and/or outputs • Versatility – our initial conditions DO NOT have to be 0

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Once we solve for i(t) => solve for other values Multiple ways of describing the same system

• Converts N-th order differential equation into N simultaneous FIRST-ORDER differential equations 13

Example 2

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In Summary State Variables: Smallest set of LINEARLY INDEPENDENT system variables s.t. state_variables(t_0) + known input (or forcing) functions COMPLETELY determines the system. # of state variables = Dimension of the State Space # of state variables = order of the original differential equation

Desired Output 15

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